Problem: Determine how many solutions exist for the system of equations. ${3x-y = 1}$ ${-3x+y = -1}$
Solution: Convert both equations to slope-intercept form: ${3x-y = 1}$ $3x{-3x} - y = 1{-3x}$ $-y = 1-3x$ $y = -1+3x$ ${y = 3x-1}$ ${-3x+y = -1}$ $-3x{+3x} + y = -1{+3x}$ $y = -1+3x$ ${y = 3x-1}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 3x-1}$ ${y = 3x-1}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${3x-y = 1}$ is also a solution of ${-3x+y = -1}$, there are infinitely many solutions.